Question

F= X^3-3X+m , determinati m pentru care polinomul f are o radacina dubla.

Answer 1

Δ=0
Δ=-4b³d+b²c²-4ac³+18abcd-27a²d²
x³-3x+m=0
a=1
b=0
c=-3
d=m
-4(-3)³-27m²=0
27×4-27m²=0
m²=4
m=2   
x³-3x+2=0
x³-x-2x+2=0
x(x²-1)-2(x-1)=0
x(x-1)(x+1)-2(x-1)=0
(x-1)[x(x+1)-2]=0
(x-1)(x²+x-2)=0
(x-1)(x²-1+x-1)=0
(x-1)[(x-1)(x+1)+(x-1)]=0
(x-1)[(x-1)(x+1+1)=0
(x-1)(x-1)(x+2)=0
(x-1)²(x+2)=0           x=1    x=-2

m=-2
x³-3x-2=0
x³-4x+x-2=0
x(x²-4)+(x-2)=0
x(x-2)(x+2)+(x-2)=0
(x-2)[x(x+2)+1]=0
(x-2)(x²+2x+1)=0
(x-2)(x²-x+x+1)=0
(x-2)[x(x+1)+(x+1)]=0
(x-2)(x+1)(x+1)=0
(x-2)(x+1)²=0           x=-1    x=2